I downloaded a book from this site, Greek grammar by William W. Goodwin. It is written that you read diphtong "eu" as "eu" in feud (so /ju/). But in an other grammar I found that you read it like /eu/.

In that other grammar is written that diphtong "ei" is written just like that, /ei/. But, in this grammar by William W. Goodwin, it is explained about it, first it was /ei/, and then /i:/. So, I'm asking, what would be better to use, or what do you prefer?

The case of ei is more complicated. In origin there were two different sounds, the way Homer pronounced them:

1) /ei/, a true dipthong, as in εἶμι, to go.2) /e:/, a long, closed eh-like sound (somewhere I saw this compared to French "nez"), as in εἰμί, to be.

So how do you know? Well, you don't. In Attic, these sounds have merged and both are pronounced /e:/. Later on, it will become /i/.

I suppose you just should just pick one pronounciation and be consistent. If you want to know more, you could read earlier threads here, and you could also read the Wikipedia article "Ancient Greek Phonology", to get started. It can get really complicated, and you'll see that people don't always agree on Classical Greek pronunciation!